Families of Young functions and limits of Orlicz norms

Abstract Given a $\sigma $ -finite measure space $(X,\mu )$ , a Young function $\Phi $ , and a one-parameter family of Young functions $\{\Psi _q\}$ , we find necessary and sufficient conditions for the associated Orlicz norms of any function $f\in L^\Phi (X,\mu )$ to satisfy $$\begin{align*}\lim_{q\rightarrow \infty}\|f\|_{L^{\Psi_q}(X,\mu)}=C\|f\|_{L^\infty(X,\mu)}. \end{align*}$$ The constant C is independent of f and depends only on the family $\{\Psi _q\}$ . Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.